Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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36 views

How to prove the convergence of a sequence satisfying $a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1$?

Assume a positive sequence $\left\{ a_{k} \right\}$ satisfying \begin{equation} a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1, k \in \mathbb{N} \end{equation} where $c_{1},c_{2},a_{1} > 0$. ...
4
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1answer
91 views

How to prove there exist distinct $a_{i}$ such $f'(a_{1})f'(a_{2})f'(a_{3})\cdots f'(a_{n})=1$

Let $f$ be a continuous map from $[0,1]$ to $R$ that is differentiable on $(0,1)$,with $f(0)=0,f(1)=1$, show that for each postive integer $n$ there exist distinct numbers $a_{1},a_{2},\cdots,a_{n}\in ...
3
votes
0answers
38 views

“composition” of “pointwise convergent sequences of functions”

Intuitively, if $f_n\to f$ as $n\to\infty$ and $g^{(n)}_i\to f_n$ as $i\to\infty$, can we get $g_j\to f$ as $j\to\infty$? Formally, Let $\{f_n\}_n$ be a sequence of functions from $\mathbb{R}^d$ ...
3
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0answers
36 views

Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
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0answers
25 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
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2answers
30 views

What can I say about the constant of a Lipschitz condition for a scaled norm?

Let's say $X$ is a vector space with inner product $\langle \cdot,\cdot\rangle$ and induced norm $\|\cdot\|$. Then for a scalar $\theta > 0$ we define $\langle \cdot,\cdot\rangle_{\theta} := ...
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0answers
36 views

why does a nonempty convex set have a nonempty interior

Suppose $A \neq \emptyset$ is a convex set in $R^n$, or generally a metric space, then is there any neat proof to show that $A$ has nonempty interior? Thanks for the help.
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0answers
42 views

On the size of rational numbers and Irrational numbers. [duplicate]

Being a high school student, It's obvious to me that there are both an infinite number of rational and irrational numbers. However I don't really see if there is more rational than irrational, ...
3
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1answer
36 views

Smooth maps preserve dimension

I stumbled over a useful consequence, that is apparently wrong for only continuous maps. Imagine $A \subset \mathbb{R}^{n-1}$ is a compact set and $F : \mathbb{R}^{n-1} \rightarrow S^{n}$ a smooth ...
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1answer
64 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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1answer
28 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
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1answer
14 views

Lipschitz maps on locally compact groups

Suppose $G$ is a locally compact second countable group. This means that there exists a proper (closed bounded sets are compact) left invariant ($d(gx,gy) = d(x,y) \ \forall g,x,y \in G$) metric on ...
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4answers
61 views

Is there a way to show the following equality without induction

I wanted to show the following equality without using induction: $$ \sum_{k=2}^n \frac{1}{k(k-1)} = \frac{n-1}{n} $$ Any hint on how to do it?
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1answer
40 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
0
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1answer
75 views

Induction proof $\sum_{k=1}^{n}\frac{1}{\sqrt{k}} \gt \sqrt{n}$

I want to show the following statement through induction for n>1: $$ \sum_{k=1}^{n}\frac{1}{\sqrt{k}} \gt \sqrt{n} $$ The induction step isn't completly working out. I should show that the last term ...
0
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1answer
50 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
0
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1answer
26 views

Are those uses of sum and product notations correct?

I wanted to use th eproduct and um notations to describe the following sequences. Are those correct? $$ 6+12+18+24=\sum_{i=1}^4 6i \\ x_{1}-x_{2}+x_{3}-x_{4}+...+x_{7}-x_{8} = \sum_{i=1}^8 ...
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0answers
18 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
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1answer
20 views

$ \exists f:\mathbb C \setminus D \to \mathbb C$ is bounded one-one holomorphic, how?

We note that there cannot exist bounded one-one holomorphic map $f:\mathbb C \setminus \{0\} \to \mathbb C.$ Put $D=\{z\in \mathbb C: |z|\leq1\}$ (closed disk). My Question: How to show there ...
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1answer
18 views

$\lim_{N \rightarrow \infty}\dfrac{1}{N^d}|E \cap \dfrac{1}{N}\mathbb{Z^d}|$ may not exist

In this page, under Remark $1$, the limit $$\lim_{N \rightarrow \infty}\dfrac{1}{N^d}|E \cap \dfrac{1}{N}\mathbb{Z^d}|$$ may not exist. Question: For $d=1$, that is, in $\mathbb{R}$, what is the ...
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0answers
8 views

baire $1$ function

Here is a new definition of Baire Class one function. Suppose that $X$ is a complete separable metric space. A function $f:X \rightarrow \mathbb{R}$ is said to be Baire class one if for any ...
1
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1answer
32 views

Need help with this question concerning compact spaces

Let the set be given like in the following manner: $$\{x_n: n\in\mathbb N\}\subset \mathbb{R^n}$$ $$l^2=\left\{\{x_{n}\}_{n=1}^{\infty}\,\Big|\, \sum_{n=1}^{\infty}|x_n|^2<\infty\right\}.$$ Prove ...
3
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1answer
462 views

Why doesn't the nested interval theorem hold for open intervals?

Why is the condition that the intervals be closed necessary? Could someone give me an example of a sequence of nonempty, bounded, nested intervals whose intersection is empty? I can't think of one, so ...
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1answer
31 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...
-2
votes
1answer
64 views

Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? [closed]

(1)Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? (2)Let $X$ be a closed connected subset of $\mathbb C$ and which has more than one element. Does ...
1
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0answers
18 views

Weak harnack type inequality

I have reached a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution ...
1
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1answer
24 views

Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point. The background So he wants to show that any symplectic form is ...
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0answers
18 views

What's the difference between Error Analysis, Statistics, and Probability?

I am interested in what people think about this question. What is the difference between error analysis, statistics, and probability? Error analysis is not a discipline that you can find in a ...
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2answers
73 views

Evaluate $\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$ [duplicate]

Let $0 < r < 1$. Compute $$\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$$ The hint is rewrite this integral as a complex line, but I still don't know how to to ...
0
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1answer
37 views

Value of $\int_C\frac{e^z}{z}dz$ with $C$ unit circle

Compute the integral $$\int_C\frac{e^z}{z}dz$$ where $C$ denotes the unit circle with positive orientation. I was thinking that let $z = e^{it}$, $dz = ie^{it}$, then the integral will become ...
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1answer
46 views

Does there exists biholomorphic map(with suitable condition) from domain to open disk?

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk). We define $f:D\to D$ as $f(z)=z,$ for all $z\in D,$ which is clearly biholomorphic. My Question is: (1) For any $z_0\in D,$ can we choose a ...
0
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1answer
62 views

Convergence of the series $\sum_{n=0}^\infty \sin(n! \pi m \sin(1))$

In this exercise I was asked to prove the convergence of the following infinite sum: $$\sum_{n=0}^\infty \sin(n! \pi m \sin(1)),$$ where $m$ denotes any integer. I don't have any idea on how to ...
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2answers
22 views

Existence of unique solution in Banach space

Let $X$ be a Banach space and let $L : X → X$ be a bounded linear operator. Are there situations where $||L||>1$ for which there is a unique solution to $x=Lx+b$? Explain your answer. My attempt: ...
0
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1answer
18 views

Dual norm of $L_p$ space

Given $R^n$ is equipped with the norm $||x|| = (\sum_{k=1}^{n} |x_k|^p)^{\frac{1}{p}}$ for some $p ≥ 1$, what is the induced norm on the conjugate (dual) space? I couldn't figure out how to prove ...
0
votes
1answer
30 views

Orthogonal projection in complex Hilbert space

Let $X$ be a complex Hilbert space, and let $T\in L(X, X)$ denote the orthogonal projection onto a closed subspace $M ⊆ X$. (a) Determine the kernel $N(T − λI)$ and the range $R(T − λI)$ of $T − λI$ ...
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1answer
27 views

Sets where functions can be discontinuous

Consider functions of the form $f: (0,1) \to \Bbb{R}$, and let $D(f)$ denote the set of points where $f$ is not continuous. Can we get a function for which $D(f)=\emptyset$? And can we get a ...
2
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3answers
65 views

Brief book on calculus to read before studying the analysis [closed]

S.E advisers, I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and ...
1
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0answers
30 views

Uniqueness of the solution of a PDE system

If I have the following PDE system: $\frac{\delta}{\delta t}x(t,r)=-\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)$ $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)-y(t,r)$ $x(0,r)=a(r), ...
0
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3answers
59 views

proof that the square of any odd number, decreased by one, is divisible by 4

I want to proof that the square of any odd number, decreased by one, is divisible by $4$. I got the following formula: $$ \frac{(2n-1)^2-1}{4} = y $$ I want to show that y is a natural number. After ...
2
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0answers
17 views

Absolute continuity via maximal operator

I'm reading an article and there is a passage that is not very clear to me. The situation is as follows: $f$ is a continuous monotonically increasing function on $[a,b]$. Define: $$ G := [x \in ...
2
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1answer
69 views

Continuous and measurable in each variable $\implies$ product measurable?

Consider a metric space $A$ with a metric $d$, and consider the measurable space $(A,\mathcal{B}(A))$ with the Borel $\sigma$-algebra generated by $d$-open sets. Let $(\Omega,\mathcal{F})$ be a ...
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2answers
23 views

If $M$ is $F$-measurable, then is it also $F'$-measurable with $F'\subset F$?

$F$ and $F'$ are $\sigma$-algebras, and $M$ is a function from $(\Omega,F)$ to $(\mathbb{R},B(\mathbb{R}))$ If this statement is true, how to reason or understand it in a simple way?
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1answer
26 views

Relation between Lebesgue measure of two sets

question in lebesgue measure: Given that $T$ is a Jordan set of positive Lebesgue measure, $l(T)>0$. If $M \subset T $ such that $l(M)=0$ where $l(\cdot)$ denote Lebesgue measure, is it true ...
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1answer
45 views

Why doesn't this work for Rudin Exercise 8 Chapter 3 series proof?

Okay so Here is the problem: If $\sum{a_n}$ converges and $b_n$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. So I can follow the long epsilon based proof and I'm good with all ...
5
votes
1answer
112 views

Is there a function $f''(0)$ exists, $f'$ is not continuous on $(-\delta,\delta)$

Is there a function $f\colon(-\delta,\delta)\to\Bbb R$ satisfying the folowing conditions(real number $ \delta\gt0$)? (i) $f$ is differentiable on $(-\delta,\delta)$; (ii) the second derivative ...
3
votes
2answers
92 views

Is $L^1(X) \cap L^2(X)$ a closed subspace of $L^2(X)$ and $L^1(X)$?

Suppose that $X$ be a locally compact Hausdorff space. Could we say that $L^1(X)\cap L^2(X)$ is closed subspace of $L^1(X)$ and $L^2(X)$?
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2answers
21 views

Problem with one of the order property proofs

I came across an exercise in an analysis book that requires me to show that if $A$ is a real number such that $0 \leq A \leq B$ for every $B>0$, then $A=0$ What I fail to understand here is if ...
1
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1answer
34 views

Question about Entire functions

Let $D=B(z_0,R)$ be the open disc centered at $z_0$ with radius $R>0$ and $f$ be a non-constant entire function. Is it true that $f$ maps the boundary $\partial D$ of $D$ into the boundary ...
3
votes
1answer
37 views

Convexity increases the “cost” of long steps

Let $V(n)$ be a non-decreasing, convex function on $\mathbb{N}$ such that $V(0)=0$, $V(1)=1$. Let $(r_i)_{i=1}^{N}$ and $(r^{\prime}_i)_{i=1}^{N^{\prime}}$, $N^{\prime} > N$, be two sequences of ...
1
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1answer
59 views

Prove $\mathbb Q( \sqrt2)$ has only two orderings

I'm having trouble showing that there are only two unique orderings of $\mathbb Q$ restricted to square root of two. I can show that the rationals are ordered, but I can't seem to figure out how to ...